Hierarchical cascading in FEM simulations of SAW devices

ABSTRACT

The present disclosure provides systems and methods for hierarchical cascading in finite element method (FEM) simulations of surface acoustic wave (SAW) devices, which offers drastically reduced memory consumption and simulation times. In some implementations, iterative hierarchical cascading may also be applied to three-dimensional simulations of SAW devices, which may otherwise be too complex for FEM simulations due to the high number of cross-sectional degrees-of-freedom involved.

RELATED APPLICATIONS

The present application claims the benefit of and priority as acontinuation-in-part to U.S. Nonprovisional patent application Ser. No.15/406,600, entitled “Hierarchical Cascading in Two-Dimensional FiniteElement Method Simulation of Acoustic Wave Filter Devices,” filed Jan.13, 2017, which claims priority to U.S. Provisional Application No.62/380,931, entitled “Hierarchical Cascading in Two-Dimensional FiniteElement Method Simulation of Acoustic Wave Filter Devices,” filed Aug.29, 2016; and also claims the benefit of and priority to U.S.Provisional Patent Application No. 62/746,937, entitled “HierarchicalCascading in FEM Simulations of SAW Devices,” filed Oct. 17, 2018; andalso claims the benefit of and priority to U.S. Provisional PatentApplication No. 62/778,168, entitled “Hierarchical Cascading in FEMSimulations of SAW Devices,” filed Dec. 11, 2018; the entirety of eachof which is incorporated by reference herein.

NOTICE OF COPYRIGHTS AND TRADE DRESS

A portion of the disclosure of this patent document contains materialwhich is subject to copyright protection. This patent document may showand/or describe matter which is or may become trade dress of the owner.The copyright and trade dress owner has no objection to the facsimilereproduction by anyone of the patent disclosure as it appears in thePatent and Trademark Office patent files or records, but otherwisereserves all copyright and trade dress rights whatsoever.

BACKGROUND

Fast development of surface acoustic wave (SAW) filters, which isbecoming ever more complicated, demands precise and universal simulationtools. The finite element method (FEM) is very attractive due to itsremarkable generality. FEM can handle arbitrary materials and crystalcuts, different electrode shapes, and structures including multiplemetal and dielectric layers. However, the application of FEM to the SAWdevices has been hampered by 1) the difficulty of modeling theeffectively semi-infinite substrate crystal, and 2) the large memoryconsumption and slow computation times. Accordingly, in practice,simulation accuracy has been limited by memory and computationconstraints, limiting the effectiveness of these techniques.

SUMMARY

The systems and methods discussed herein provide for hierarchicalcascading in FEM simulations of SAW devices, which offers drasticallyreduced memory consumption and simulation times. In someimplementations, iterative hierarchical cascading may also be applied tothree-dimensional simulations of SAW devices, which may otherwise be toocomplex for FEM simulations due to the high number of cross-sectionaldegrees-of-freedom involved.

Optional features of one aspect may be combined with any other aspect.

BRIEF DESCRIPTION OF THE DRAWINGS

The patent or application file contains at least one drawing executed incolor. Copies of this patent or patent application publication withcolor drawings will be provided by the Office upon request and paymentof the necessary fee.

The details of one or more implementations are set forth in theaccompanying drawings and the description below. Other features,aspects, and advantages of the disclosure will become apparent from thedescription, the drawings, and the claims, in which:

FIG. 1A is a model of an implementation of a surface acoustic wavedevice;

FIG. 1B is an abstracted illustration of the implementation of thesurface acoustic wave device of FIG. 1A;

FIG. 1C is a modified illustration of the abstraction of FIG. 1Brepresenting the use of an implementation of a hierarchical cascadingfinite element analysis method on the surface acoustic wave device ofFIG. 1A;

FIGS. 2A and 2B are illustrations of an example of a unit block and acomputational mesh of a unit block, for implementations of hierarchicalcascading finite element analysis;

FIG. 3 is a graph of slowness curves for bulk acoustic waves in animplementation of a bulk acoustic wave device;

FIG. 4A is an illustration of a series of computational meshes of unitblocks, for implementations of hierarchical cascading finite elementanalysis;

FIG. 4B is an illustration of conversion of a computational mesh of aunit block into a multi-port model, according to some implementations;

FIG. 4C is an illustration of cascading multi-port models to model acombination of unit blocks, according to some implementations;

FIG. 4D is an illustration of the series of computational meshes of unitblocks of FIG. 4A, after a first iteration of a hierarchical cascadingprocess, according to some implementations;

FIGS. 4E and 4F are illustrations of an iterative hierarchical cascadingprocess through multiple iterations, according to some implementations;

FIG. 4G is an illustration of a hierarchical cascading tree for asynchronous resonator, according to some implementations;

FIGS. 5A-5C are illustrations of two and three dimensional models of anelectrode array, according to some implementations;

FIG. 6A is a flow chart of an implementation of a method forhierarchical cascading analysis;

FIG. 6B is a flow chart of an implementation of a method for iterativehierarchical cascading analysis;

FIG. 7A illustrates graphs of admittance over frequency for a measuredexample surface acoustic wave resonator and a simulated surface acousticwave resonator, according to some implementations;

FIG. 7B illustrates shear-horizontal polarization at selectedfrequencies for the example surface acoustic wave resonator of FIG. 7A,according to some implementations;

FIG. 8 illustrates graphs of admittance over frequency for a simulatedtemperature compensated surface acoustic wave resonator at differentboundary conditions, according to some implementations;

FIG. 9 is a graph of Q-factor over frequency for a simulated temperaturecompensated surface acoustic wave resonator, according to someimplementations;

FIG. 10 is an illustration of a decomposition of an array into aplurality of unique unit blocks, according to some implementations;

FIG. 11 is an illustration of slowness curves for a linear surfaceacoustic wave device, according to some implementations;

FIG. 12 illustrates graphs of admittance over frequency for examplemodels of a surface acoustic wave device generated via implementationsof direct and iterative hierarchical cascading FEM in two and threedimensions;

FIG. 13 illustrates example displacement profiles for the surfaceacoustic wave device of FIG. 12, according to some implementations;

FIG. 14 is an example graph illustrating relative error in estimatedharmonic admittance via iterative hierarchical cascading FEM overiteration count, according to some implementations;

FIG. 15 is a block diagram of an implementation of a system forhierarchical cascading in FEM simulations and for designing electronicfilters;

FIG. 16 is a block diagram of a wireless telecommunications system,according to some implementations

FIG. 17 is a flow diagram illustrating the design and construction of afilter using the FEM hierarchical cascading technique, according to someimplementations; and

FIG. 18 is a block diagram of a system for hierarchical cascading in FEMsimulations and for designing electronic filters, according to someimplementations.

Like reference numbers and designations in the various drawings indicatelike elements.

DETAILED DESCRIPTION

The systems and methods discussed are directed to hierarchical cascadingand iterative hierarchical cascading, techniques to greatly reduce thecomputational time and memory requirements for finite element modeling(FEM) of surface acoustic wave (SAW) devices.

Fast development of SAW filters, which is becoming ever morecomplicated, demands precise and universal simulation tools. FEM is veryattractive as a simulation and analysis technique due to its remarkablegenerality. FEM can handle arbitrary materials and crystal cuts,different electrode shapes, and structures including multiple metal anddielectric layers. However, the application of FEM to the SAW deviceshas been hampered by 1) the difficulty of modeling the effectivelysemi-infinite substrate crystal, and 2) the large memory consumption andslow computation times. Accordingly, in practice, simulation accuracyhas been limited by memory and computation constraints, limiting theeffectiveness of these techniques.

In many implementations, a perfectly matched layer (PML) technique maybe used to model the substrate crystal. Certain piezo-electric materialswith unsuitable anisotropy are prone to instabilities in implementationsusing a convolution stretched-coordinate PML (C-PML) approach, and theyremain difficult to simulate. Unfortunately, this anomalic categoryincludes several technologically important SAW substrates, such as 128°YX-cut LiNbO₃ in two dimensions, and 42° YX-cut LiTaO₃ in threedimensions. A multi-axial PML (M-PML) technique may be used in someimplementations to provide an acceptable, albeit less efficientsolution.

Referring briefly to FIG. 1A, illustrated is a model of animplementation of a SAW device 100. In non-hierarchical FEM analysis,the simulation covers the entire geometry of the device (open boundaryconditions are mimicked by surrounding the simulation domain with anartificial computational material or PML, which absorbs the wavesincident to the layer). A practical FEM model of the regions around asingle electrode may include thousands of finite elements with hundredsof variables, resulting in tens or hundreds of thousands of potentialvariables or degrees-of-freedom; a FEM model of a complete SAW deviceswith hundreds of electrodes can include millions of potential variables,impractical for use in even the fastest computing systems.

FIG. 1B is an abstracted illustration 100′ (not to scale) of theimplementation of the SAW device of FIG. 1A. The SAW device 100′, whichmay be a filter, resonator, coupling element, or any other such device,may comprise a finite array finite array of conductive electrodes 110(e.g., the interdigitated metal fingers of a resonator) and asemi-infinite substrate crystal 108. One or more dielectric orpiezoelectric layers below, above, and between the electrodes 110, mayalso be modeled. The SAW structure 100′ may also comprise a passivationlayer (e.g., SiO₂) 112 disposed over the electrodes 110 and substrate108. A regional domain consisting of the electrodes 110, a portion ofthe substrate 108 adjacent the electrodes 110, and a portion of a vacuum106 above the electrodes 110 may be defined and modeled in fine detailvia a mesh. The regional domain may be surrounded by an artificialcomputational material consisting of a substrate PML 104 and a vacuumPML 102, which interfaces smoothly with the modeled domain and which hasthe property that incident propagating acoustic waves are converted intoexponentially decaying acoustic waves. The regional domain, along withthe substrate PML 104 and vacuum PML 102, are computationally meshed anda frequency response is computed using the FEM. While this approach isunstable in some substrates with unsuitable anisotropy, in manyimplementations, using a PML is a very efficient solution to the openboundary problem.

Although the use of PMLs addresses the open boundary condition problem,the second problem of addressing the large number of degrees-of-freedomremains. A practical FEM model of the regions around a single electrodemay contain thousands of finite elements on the order of 1,000-10,000unknown variables. Thus, the FEM model of a complete SAW structure withseveral hundreds of electrodes can be huge; up to millions of equations.

This problem of memory consumption and computation times may beaddressed through the use of hierarchical cascading algorithms that takeadvantage of the periodic structure typical of SAW devices. FIG. 1C is amodified illustration of the abstraction of FIG. 1B representing the useof an implementation of a hierarchical cascading finite element analysismethod on the SAW device of FIG. 1A. Rather than modeling the entire SAWdevice, the SAW device may be partitioned into portions or unit blocks115B, of which many may be identical. Only unique unit blocks need bemodeled or simulated, while identical unit blocks may be excluded frommodeling. The sides of the device may be terminated in dedicated PMLblocks 115A, 115C for analysis, as external degrees-of-freedom. Thedevice 100′ is accordingly analyzed as a hierarchical tree of cascadingoperations, where adjacent smaller blocks are combined into largerblocks, eventually covering the whole device. In 2D this is drasticallymore efficient than conventional FEM, and may also be applied in manyimplementations to 3D SAW simulation and bulk acoustic wave (BAW)devices.

FEM Modeling of Unit Blocks

In the hierarchical cascading method, the device geometry is partitionedinto a sequence of repeating unit blocks 200 (e.g. similar to unit block115B discussed above), an example implementation of which is illustratedin FIG. 2A. As shown, a typical unit block 200 may include a rectangularsubstrate volume 202, an electrode 204, and the vacuum 206 above. Theelectrode 204 may consist of multiple metals, depending onimplementation, and may be covered by a stack of dielectric layers(yellow and red in the illustration of FIGS. 2A and 2B). Γ_(L) and Γ_(R)denote the left and right edges of the unit block 200, respectively,while Γ_(e) is the electrode boundary. The acoustic aperture dimensionis into the page. FIG. 2B illustrates an implementation of an FEM mesh200′ of the unit block 200, using triangular and quadrilateral elements.Dense meshing around the electrode may improve the accuracy of simulatedsurface charge distribution, in some implementations. The top and bottomPML are not shown in full height, and may extend farther in manyimplementations. Accordingly, the unit blocks 200 are subdivided intofinite elements, yielding the computational mesh 200′ of the geometry,shown in FIG. 2B.

A. FEM System Equation

Consider a unit block under harmonic excitation at angular frequencyω=2πf. Non-hierarchical FEM analysis yields a linear system ofequations:[K+iωD−ω ² M](x)=(F)  (eq. 1)Here, the expression in the brackets is the system matrix, consisting ofthe stiffness matrix K, damping matrix D, and mass matrix M. They areinherently symmetric. The vector x contains the DOFs of the model: thenodal values of mechanical displacement and electric potential at thenodes. The vector F contains the external sources—the charge density andthe boundary stresses.

The degrees-of-freedom and the external sources can be classified intothose associated with the left edge (L), the right edge (R), interior(I), and the electric potential connected to an electrode (v).Correspondingly, the system of equations (1) can be reordered as:

$\begin{matrix}{{\begin{bmatrix}A_{LL} & A_{LI} & 0 & A_{LV} \\A_{IL} & A_{II} & A_{IR} & A_{IV} \\0 & A_{RI} & A_{RR} & A_{RV} \\A_{VL} & A_{VI} & A_{VR} & A_{VV}\end{bmatrix}\begin{pmatrix}X_{L} \\X_{I} \\X_{R} \\v\end{pmatrix}} = \begin{pmatrix}\tau_{L} \\\tau_{S} \\\tau_{R} \\{- q}\end{pmatrix}} & \left( {{Eq}.\mspace{11mu} 2} \right)\end{matrix}$Here, A is the reordered system matrix. The equations associated withthe electric DOF on the electrode have also been included; this can beinterpreted as integration over charge density. On the right-hand side,τL and τR are integrals over surface stresses at the left and rightedge. These will cancel out in the cascading process. In most use cases,interior stresses τs=0. The scalar q denotes the net surface charge atthe electrode boundary; the current flowing into the electrode isI=iωWq. If the unit block has no electrode, the related components arefilled with zeros.B. Perfectly Matched Layers

Perfectly matched layers (PML) are used to mimic open boundaryconditions. An ideal PML absorbs all incident acoustic radiation withoutreflections. In implementations of hierarchical cascading analysis, eachunit block may be conceptualized or modeled as including a bottom PMLand a vacuum PML, absorbing energy in the vertical direction. Inaddition, the dedicated PML blocks at the sides also absorb radiation inthe horizontal direction, as shown above in units 115A and 115C of FIG.1B. Depending on the anisotropy of the substrate, the differentdirections may require using different PML techniques, with differenttypes coexisting in the corners. In various implementations, one or moreof the following techniques may be employed.

1) Convolution Stretched-Coordinate PML:

The C-PML technique is particularly well suited for elastic andpiezoelectric problems. Within the PML, the physical coordinates (x₁,x₃) are replaced with complex-valued stretched coordinates of the form:

$\begin{matrix}\left\{ \begin{matrix}{{{\overset{\sim}{x}}_{1} = {x_{1} - {i{\int{{\mu_{1}\left( x_{1} \right)}{dx}_{1}}}}}},} \\{{\overset{\sim}{x}}_{3} = {x_{3} - {i{\int{{\mu_{3}\left( x_{3} \right)}{{dx}_{3}.}}}}}}\end{matrix} \right. & \left( {{Eq}.\mspace{11mu} 3} \right)\end{matrix}$Generally, the stretching factors may be complex-valued andfrequency-dependent. Stretching is only applied in the direction whereabsorption is required, in many implementations.

To demonstrate how the C-PML works, consider the impact of a bottom PMLon a downward-propagating wave:e ^(ik) ³ ^(x) ³ →e ^(ik) ³ ^(x) ³ e ^(k) ³ ^(∫μ) ³ ^((x) ³ ^()dx) ³  (Eq. 4)With Re(μ₃)>0, the propagating wave is effectively converted into adecaying wave, which tends to zero as x₃→−∞. Moreover, any residual wavecomponent reflected from the bottom boundary will further decay on itsway back upwards to the surface. An imaginary part Im(μ₃)>0 can beinterpreted as geometric scaling; it enhances the decay of surface modesinto the PML. However, it also accelerates the oscillation ofpropagating waves.

In FEM implementation, the coordinate stretching corresponds to thesubstitution:

$\begin{matrix}\left. \frac{\partial}{\partial x_{k}}\rightarrow{\left( \frac{1}{1 + {i\;\mu_{k}}} \right)\frac{\partial}{\partial x_{k}}} \right. & \left( {{Eq}.\mspace{11mu} 5} \right)\end{matrix}$in the field equations. It can be implemented directly in elementintegration routines of the FEM code, or be subsumed in the materialsconstants.

In theory, the surface impedance of the PML is identical to that of thenormal substrate: the layer is perfectly matched. In practical FEMimplementation, with the differential equations approximated withdiscrete equations, the matching is not perfect, especially for waveswith a shallow propagation angle. To minimize reflections, thestretching factors and the other properties of the PML must be chosenwith great care. A common practice is to choose an absorption profilewhich vanishes at the interface between the normal substrate and thePML. One such choice is:

$\begin{matrix}{{\mu_{i}(\xi)} = \left\{ \begin{matrix}0 & {\xi < \xi_{\min}} \\{\sin^{2}\left( {\frac{\pi}{2}\frac{\xi - \xi_{\min}}{\xi_{\max} - \xi_{\min}}} \right)} & {\xi \in \left\lbrack {\xi_{\min},\xi_{\max}} \right\rbrack}\end{matrix} \right.} & \left( {{Eq}.\mspace{11mu} 6} \right)\end{matrix}$Here, ξ denotes the relevant coordinate direction, and μ_(i,max) is themaximum stretching factor.

The stretching factors can be chosen inversely proportional tofrequency. This makes the attenuation rate within the PML independent ofthe frequency, which is particularly handy in time-domain simulations.However, the frequency-independent stretching factors used in Eqs.(3)-(6) also have a significant advantage: the matrices (K, D, M) in Eq.(1) are frequency-independent and need to be evaluated only once.

In many implementations, the C-PML technique cannot absorb waves withnegative phase velocity in the direction of the PML. This situation mayoccur in substrates with unfavorable shape of the slowness curves. As anexample, FIG. 3 is a graph of slowness curves for bulk acoustic waves(BAWs) in an implementation of a 128° YX-cut LiNbO₃ device. Blue, red,and magenta curves correspond to slow shear, fast shear, andlongitudinal bulk-acoustic waves, respectively. For s_(x1)∈(2.104 . . .2.151)·10⁻⁴, the fast shear wave has concave slowness curvature. Thearrows indicate the direction of power flow, parallel to the outwardnormal of the slowness curves.

Consider bulk waves propagating in a semi-infinite substrate x₃<0. Thephysically relevant solutions are those with zero or negative power flowalong x₃. On the concave region of the slowness curves, these includefast shear waves with upward phase velocity (s_(x3)>0). Substitution ofsuch modes in Eq. (4) results in exponential amplification, not indecay. In practical implementation, the problem manifests as instabilityof the PML.

2) Multi Axial Perfectly Matched Layer:

In some implementations, multi-axial PML may be used to improveinstability issues in the simulation. In contrast to C-PML, wherecoordinate stretching is applied only in the direction of the layer, inM-PML coordinate stretching is applied also parallel to the layerinterface. The ratio of tangential and normal stretching factors is keptconstant; for example, in a bottom PML, μ₁(x₃)=rμ₃(x₃)≠0. Theproportionality constant r may be chosen based on the anisotropy of thesubstrate and the direction of the PML. For r→0, the M-PML reduces to aconventional C-PML.

A sufficiently large stretching ratio r stabilizes the M-PML.However—contrary to its name—the layer is not perfectly matched to thesubstrate. In many implementations, the M-PML technique is more prone toreflections than the C-PML technique, especially for shallow propagationangles. The computational mesh, the r-parameter, and the attenuationprofile μ(ξ) should be optimized simultaneously, carefully consideringthe trade-off between the absorbing efficiency of the M-PML and thestrength of reflections.

3) Hierarchically Cascaded PML:

In some implementations of analysis of BAW devices, long damping layersimplemented with the hierarchical cascading method can be usedeffectively as side PMLs. With longer layers, lower attenuation ratescan be used, reducing matching problems and, as coordinate stretching isnot used, there are no stability problems. However, the idea also workswith coordinate stretching. Hierarchical cascading makes long C-PMLs orM-PMLs with piecewise flat absorption profile computationallyattractive.

4) Anisotropic Perfectly Matched Layer for Vacuum:

Neither C-PML nor M-PML techniques work in vacuum. Instead, in someimplementations, a strong artificial anisotropy is introduced to thedielectric permittivity of vacuum ϵ₀ as follows:

$\begin{matrix}\left. \epsilon_{0}\rightarrow{\epsilon_{0}\begin{bmatrix}{\mu_{vacuum}\left( x_{3} \right)} & 0 \\0 & {\mu_{vacuum}^{- 1}\left( x_{3} \right)}\end{bmatrix}} \right. & \left( {{Eq}.\mspace{11mu} 7} \right)\end{matrix}$Here, the anisotropy profile μ_(vacuum)(x₃) is a parameter. To avoidnumerical problems, it starts from μ_(vacuum)=1, but increases deeperinto the PML, in such manner that the normal component of the relativepermittivity becomes much less than unity. As a result, analysis atvacuum is both extremely efficient and easy to implement.

Hierarchical Cascading Method

A. From FEM Model to B-Matrix

In absence of interior stresses (τs=0), the internal DOFs x_(I) can beeliminated from the matrix of Eq. (2) by forming the Schur complement ofA_(II). This results in a new system of equations, where the onlyvariables are the electric potential and the DOFs associated with thenodes at the left- and right-hand side interfaces:

$\begin{matrix}{{\begin{bmatrix}B_{11} & B_{12} & B_{13} \\B_{21} & B_{22} & B_{23} \\B_{31} & B_{32} & B_{33}\end{bmatrix}\begin{pmatrix}X_{L} \\X_{R} \\v\end{pmatrix}} = \begin{pmatrix}\tau_{L} \\\tau_{R} \\{- q}\end{pmatrix}} & \left( {{Eq}.\mspace{11mu} 8} \right)\end{matrix}$The 3×3 matrix in the above equation may be referred to in someimplementations as the “B-matrix”. The FEM system matrix A in Eq. (2) issymmetric and very sparse. The B-matrix is also symmetric but full. Itshares some similarity with the admittance matrices in network theory.B. Electric Connectivity

An extended B-matrix is needed to cover all electric ports of a SAWdevice and to connect each electrode to the correct potential. Let thevector V≡(ν₁ . . . ν_(K))^(T) contain all the K potentials present inthe device, and collect the corresponding T net surface charges to Q≡(q₁. . . q_(K))^(T). Consider unit block A. Connectivity vector Γ^(A) isdefined as the K×1 vector:

$\begin{matrix}{\Gamma_{j}^{A} = \left\{ \begin{matrix}1 & {{{if}\mspace{14mu}{electrode}\mspace{14mu}{in}\mspace{14mu} A\mspace{14mu}{is}\mspace{14mu}{connected}\mspace{14mu}{to}\mspace{14mu} v_{j}},} \\0 & {{otherwise}.}\end{matrix} \right.} & \left( {{Eq}.\mspace{11mu} 9} \right)\end{matrix}$This vector may be used to compute an extended B-matrix, defined as:

$\begin{matrix}{{\begin{bmatrix}B_{11} & B_{12} & {B_{13}\Gamma^{T}} \\B_{21} & B_{22} & {B_{23}\Gamma^{T}} \\{\Gamma\; B_{31}} & {\Gamma\; B_{32}} & {\Gamma\; B_{33}\Gamma^{T}}\end{bmatrix}\begin{pmatrix}X_{L} \\X_{R} \\V\end{pmatrix}} = \begin{pmatrix}\tau_{L} \\\tau_{R} \\{- Q}\end{pmatrix}} & \left( {{Eq}.\mspace{11mu} 10} \right)\end{matrix}$The extended B-matrix enables correct handing of N-port devices in thecascading process. It remains symmetric. The effect of electroderesistivity can also be subsumed to the extended B-matrix.C. Cascading Two B-Matrices

Let A and B be two adjacent blocks, with compatible meshes at the sharedinterface A∩B. The respective B-matrices satisfy:

$\begin{matrix}{{\begin{bmatrix}B_{11}^{A} & B_{12}^{A} & B_{13}^{A} \\B_{21}^{A} & B_{22}^{A} & B_{23}^{A} \\B_{31}^{A} & B_{32}^{A} & B_{33}^{A}\end{bmatrix}\begin{pmatrix}X_{L} \\X_{R}^{A} \\V\end{pmatrix}} = \begin{pmatrix}\tau_{L} \\\tau_{R}^{A} \\{- Q^{A}}\end{pmatrix}} & \left( {{Eq}.\mspace{11mu} 11} \right) \\{{\begin{bmatrix}B_{11}^{B} & B_{12}^{B} & B_{13}^{B} \\B_{21}^{B} & B_{22}^{B} & B_{23}^{B} \\B_{31}^{B} & B_{32}^{B} & B_{33}^{B}\end{bmatrix}\begin{pmatrix}X_{L}^{B} \\X_{R} \\V\end{pmatrix}} = \begin{pmatrix}\tau_{L}^{B} \\\tau_{R} \\{- Q^{B}}\end{pmatrix}} & \left( {{Eq}.\mspace{11mu} 12} \right)\end{matrix}$The mechanical stresses, and normal electric displacement should becontinuous across the shared edge: X_(R) ^(A)=X_(L) ^(B)≡X_(C), andτ_(R) ^(A)=−τ_(L) ^(B). Consequently, X_(C) can be eliminated via:

$\begin{matrix}{X_{C} = {{- {\left\lbrack {B_{22}^{A} + B_{11}^{B}} \right\rbrack^{- 1}\left\lbrack {{B_{21}^{A}\mspace{14mu} B_{12}^{B}\mspace{14mu} B_{23}^{A}} + B_{13}^{B}} \right\rbrack}}\begin{pmatrix}X_{L} \\X_{R} \\V\end{pmatrix}}} & \left( {{Eq}.\mspace{11mu} 13} \right)\end{matrix}$

Back substitution of Eq. (13) into Eq. (11) and Eq. (12) yields:

$\begin{matrix}{{\begin{bmatrix}B_{11}^{AB} & B_{12}^{AB} & B_{13}^{AB} \\B_{21}^{AB} & B_{22}^{AB} & B_{23}^{AB} \\B_{31}^{AB} & B_{32}^{AB} & B_{33}^{AB}\end{bmatrix}\begin{pmatrix}X_{L} \\X_{R} \\V\end{pmatrix}} = \begin{pmatrix}\tau_{L} \\\tau_{R} \\{- Q^{AB}}\end{pmatrix}} & \left( {{Eq}.\mspace{11mu} 14} \right)\end{matrix}$where Q^(AB)≡Q^(A)+Q^(B). The cascaded B-matrix in Eq. (14) fullydescribes the response of the combined block. It is also symmetric.

The size of a B-matrix depends only on the number of DOFs at the edgesand on the number of electric connections. If all mesh edges arecompatible, the cascaded B-matrix has the same size as the originalB-matrices. Moreover, cascaded B-matrices can be further cascaded todescribe longer structures. Hence, in many implementations ofhierarchical cascading, a B-matrix can describe SAW structures from asingle unit block to aggregated sequences of arbitrarily manyelectrodes, providing easy and efficient scalability.

D. Hierarchical Cascading

In implementations of hierarchical cascading, the SAW structure may bedescribed as a series of cascading operations. At the lowest level thedevice is decomposed into unit blocks, which typically contain only oneelectrode or none at all. The device structure is analyzed automaticallyto identify repeated patterns at different length scales; the aim is touse as few cascading operations as possible. This is somewhat analogousto text compression algorithms.

FIG. 4A is an illustration of a series 400 of computational meshes ofunit blocks 402, for implementations of hierarchical cascading finiteelement analysis. As discussed above, a SAW device may be conceptuallypartitioned into a series of unit blocks 402. The unit blocks may besubstantially identical, in many implementations, or may be grouped intoidentical subsets (e.g. unit blocks having a ground connection to anelectrode, unit blocks having a positive voltage connection to anelectrode, etc.).

As discussed above, each unit block may be modeled as a singlemulti-port device, with each port having many degrees-of-freedom. FIG.4B is an illustration of conversion of a computational mesh of a unitblock 402 into a multi-port model 402′, according to someimplementations. As shown, the resulting model of the unit block may berepresented by a B-matrix as discussed above.

FIG. 4C is an illustration of cascading multi-port models 402′ to modela combination of unit blocks, according to some implementations. Asshown, adjacent multi-port models 402A′, 402B′ may be combined (e.g. viaEqs. 11-14) to generate a combined unit block model 404′, which maysimilarly comprise a single multi-port device model with each porthaving degrees-of-freedom representing the combined contribution of eachunit block 402′ contained within the combination. The combined B-matrixB_(AB) of the combined unit block model 404′ is of the same size asmatrices B_(A) and B_(B).

Similarly, FIG. 4D is an illustration of the series of computationalmeshes of unit blocks of FIG. 4A, after a first iteration of ahierarchical cascading process, according to some implementations. Asshown, individual unit block models 402′ may be combined into a seriesof combined unit block models 404′, each represented with a dashedoutline. As shown, a single iteration has occurred, creating a secondlevel of the hierarchy. The process may be repeated to further combinealready-combined models into larger and larger blocks, until the entiredevice may be modeled as a B-matrix, as shown in the illustrations ofFIGS. 4E and 4F. For example, having modeled two adjacent combinationunit blocks, each having a two unit block width, the combination unitblocks may be further combined to generate a single B-matrix model ofall four individual unit blocks, with the process further iteratinguntil a single B-matrix model represents the entire device. The fullsolution may be recovered from the single B-matrix model with inversecascading, as discussed herein, to recover the originaldegrees-of-freedom.

Furthermore, not every block needs to be explicitly analyzed and itsmatrix calculated. In practice, many blocks or sets of adjacent blocksmay be repeated throughout the device or may be identical to otherblocks within the device. Accordingly, once modeled, the B-matrix forthe block or combination may be re-used for other identical blocks orcombinations within the device. An illustrated example of how thehierarchical cascading works is shown in the illustration of FIG. 4G. Aresonator 420 may comprise a series of identical electrodes withdifferent connectivity, represented by the values 0 (grounded), 1 (upperbus), and 2 (lower bus). Each node in the tree represents a B-matrix forthe electrode sequence in the box. Initially at step 430, the system istoo complex to be modeled directly. Instead, through cascading of steps430-440, the unit blocks may be divided down into identical sets ofrepresentative blocks which may then be analyzed, and then reconstructedto generate the complete model.

For example, at step 430 unique unit blocks 422 (e.g. 0, 1, and 2, shownin black) may be analyzed. Block 424 may be skipped, as it is identicalto a block that has already been analyzed. The combination of each pairof unit blocks may be used to generate the four unit blocks at step 432.The blocks shown in white (e.g. 00 and 12) may be evaluated as acombination of the two already analyzed unit blocks that make them up.The blocks shown in grey are identical to the blocks in white, and neednot be analyzed. Similarly, at steps 434, 436, and 438, larger blocksmay be constructed as a combination of previously evaluated blocksand/or may be identical to already constructed blocks. As a result, viathe ten cascading operations shown, the model of the device 420 may bedecomposed into a structure that may be more efficiently analyzed, andcascaded to generate the complete model.

E. Solving the Cascaded System

The outcome of the cascading process is a single B-matrix, whichdescribes the whole structure. Finally, the structure is terminated withside PMLs and solved for external electric excitation. Assuming that theside PMLs absorb all incident acoustic radiation, zero displacement canbe assumed at the exterior side. For the PML at the left side (LPML):[B ₂₂ ^(LPML)](x _(R) ^(PML))=(τ_(R) ^(PML))  (Eq. 15)and likewise for the PML at the right side (RPML). Cascading the deviceB-matrix with the PML blocks yields:B ₁₁ ^(tot) =B ₁₁ +B ₂₂ ^(LPML)  (Eq. 16)andB ₂₂ ^(tot) =B ₂₂ +B ₁₁ ^(RPML)  (Eq. 17)

Let there be N_(port) electric voltages, and let e be the 1×N_(port)matrix with all elements being unity. The desired voltage excitationsare stated, relative to an unknown reference potential V_(ref), asfollows:V=ΔV+e ^(T) V _(ref)  (Eq. 18)

Charge neutrality is enforced by requiring that the sum over all surfacecharges vanishes, i.e., eQ=0. After these substitutions, the finalsystem of equations takes the form:

$\begin{matrix}{{\begin{bmatrix}B_{11}^{tot} & B_{12} & {B_{13}e^{T}} \\B_{21} & B_{22}^{tot} & {B_{23}e^{T}} \\{eB}_{31} & {eB}_{32} & {{eB}_{33}e^{T}}\end{bmatrix}\begin{pmatrix}X_{L} \\X_{R} \\V_{ref}\end{pmatrix}} = {- \begin{pmatrix}B_{13} \\B_{23} \\{eB}_{33}\end{pmatrix}}} & \left( {{Eq}.\mspace{11mu} 19} \right)\end{matrix}$The net electric currents can be directly evaluated asI=−iωW(B ₃₁ X _(L) +B ₃₂ X _(R) +B ₃₃ V)  (Eq. 20)The other observables of interest can be calculated as a post-processingstep.F. Postprocessing

If desired, all internal degrees-of-freedom can be retrieved byinverting the cascading process shown in FIG. 4G. In suchimplementations, starting from the known top-level solution (x_(L),x_(R), V), the hierarchical cascading tree may be walked downwards toresolve variables at block interfaces. The B-matrices present at eachlevel of the hierarchy are split into their respective constituents,using Eq. (13) above to solve the variables at the shared edge. Theprocess is continued until the variables at all block edges have beenretrieved. The internal degrees-of-freedom x_(I) in each copy of theunit block can be retrieved from Eq. (2).

The knowledge of the full FEM solution enables evaluation andvisualization of various observables-of-interest. These observables mayinclude current distribution in the electrodes, mechanical displacementdistribution, and power flow distribution. In Q-factor analysis, thestored electroacoustic energy, resistive losses, and losses due toacoustic radiation can be evaluated independently.

Referring briefly ahead to FIG. 6A, a flow chart of an implementation ofthe above process is illustrated. The process may sometimes be referredto as direct hierarchical cascading, in contradistinction to iterativehierarchical cascading, discussed below in connection with FIG. 6B.Still referring to FIG. 6A and in brief overview, at step 602, coreblock FEM models may be identified, and at step 604, B-matrices for thecore block models may be evaluated or solved. At step 606, a cascadingprocess may be performed to successfully evaluate larger unit blocksbased on combinations of analyzed unit blocks. In some implementationsor optionally, an inverse cascading process may be utilized at step 608(shown in dashed line) to identify all internal degrees-of-freedom.Finally, post processing steps may be applied to finalize the model atstep 610.

Still referring to FIG. 6A and in more detail, at step 602, a physicalmodel of the AW structure is defined and partitioned into a plurality oforiginal unit blocks. Each original unit block may comprise zero or oneelectrode in many implementations. At least one core block is identifiedwithin the plurality of original unit blocks, and an FEM analysis isperformed to compute the electrical characteristics of the core block(s)and produce a sparse, symmetric FEM system matrix, A, for each coreblock. In some implementations, the FEM is used to compute the DOFs inthe form of acoustic and electric fields inside of the core block(s)excited by the electric potential (if any) on the electrodes (if any)within the core block(s) and the forces and electric potentialsoccurring at the boundaries of the core block(s).

At step 604, the internal DOFs are removed from each of the meshed coreblock(s) to produce reduced system B-matrices or dense, symmetric“boundary matrices” representing reduced core blocks. Computing theresponse of each of the core blocks using the FEM thus may includegenerating an A-matrix having left-side boundary DOFs, right-sideboundary DOFs, and internal DOFs, and removing the internal DOFs fromthe A-matrix to generate a B-matrix comprising only the left-sideboundary DOFs, the right-side boundary DOFs, and the electric potentialand net surface charge on the electrode(s) (if any), as discussed above.In addition, in some implementations, electrode resistive losses can besubsumed into the B-matrices. In some implementations, specifiedelectrical connections may be employed to form extended B-matrices thatdefine one or more types of original unit blocks from each core-blockB-matrix (e.g. 0, 1, 2 as discussed above in connection with FIG. 4). Insome implementations, the number of original unit blocks may be reduceddown to a minimum number of types (each type with its own uniquevoltage), so that the unit blocks can be more efficiently cascaded downto a single block.

After the types of original unit blocks have been determined from boththe core block(s) and the electrical connections of the original unitblocks that the respective core block(s) physically represent, ahierarchical cascading pattern is determined from the nature and patternof the original unit blocks at step 606, and adjacent sets of unitblocks originating from the unit blocks (e.g. blocks 0, 1, 2 of FIG. 4,as shown) are either combined or transferred from their currenthierarchical level to the next in accordance with the determinedhierarchical cascading pattern until a single block subsuming all of theoriginal unit blocks is realized. A set of adjacent blocks may, e.g.,include only original unit blocks, at least one original unit block andat least one cascaded unit block, or only cascaded unit blocks.

In particular, unique sets of adjacent original unit blocks and/orcascaded unit blocks, together with any “orphaned” original unit blocksor cascaded unit blocks, are identified at the current hierarchicallevel, and the responses (electrical characteristics) of each adjacentblock set at this current hierarchical level are cascaded (combined) todetermine the responses of larger blocks at the next hierarchical leveland the “orphaned” blocks at this hierarchical level are simplytransferred to the next hierarchical level.

In some implementations, cascading the responses of each set of adjacentunit blocks (either original or previously cascaded) may includecombining the extended and/or cascaded B-matrices of the respective setof adjacent unit blocks into a single new combined C-matrix havingleft-side boundary DOFs corresponding to the left-side boundary DOFs ofa left one of the respective set of adjacent unit blocks, right-sideboundary DOFs corresponding to a right one of the respective set ofadjacent unit blocks, and internal DOFs (center DOFs in the case wherethe set of adjacent unit blocks only include two unit blocks)corresponding to shared edges or a shared edge between adjacent ones ofthe unit blocks; and removing the internal DOFs from the single newcombined C-matrix to create a new cascaded B-matrix comprising onlyleft-side boundary DOFs and right-side boundary DOFs. In someimplementations, once the characteristics of an original unit block orcascaded unit block are computed, such computed original unit blocks orcascaded unit blocks can be conveniently referenced during subsequentcascading operations at the same hierarchical level or at the nexthierarchical level. The cascading process may repeat iteratively foreach subsequent hierarchical level until the FEM hierarchical cascadingprocess has resulted in a single block subsuming all of the originalunit blocks.

Iterative Hierarchical Cascading

While hierarchical cascading is highly efficient in 2D simulations, thebenefits of the approach may be somewhat reduced in 3D, in someimplementations. Consider three different simulation types: 2D finite,3D periodic, and 3D finite structure simulation. As a metric of thememory consumption, consider the amount of RAM required to store asingle B-matrix in a high-accuracy “stress test” case. Likewise, as agauge of the simulation speed, consider the computational complexity ofcascading two B-matrices.

An example implementation of a unit cell is depicted in FIGS. 5A-5C,including an electrode pair, a gap, busbars, the underlyingpiezoelectric substrate mesh, the vacuum above, and side PMLs (notshown). FIG. 5A illustrates the unit cell in a 2D finite mode, with 16nodes and 200 boundary degrees-of-freedom, resulting in 3200 interiordegrees-of-freedom. FIG. 5B illustrates the unit cell in a 3D periodicmode, with 32 nodes and 200 vertical degrees-of-freedom, resulting in6400 boundary degrees-of-freedom. In periodic simulationimplementations, periodic boundary conditions are applied along thex-direction. Hierarchical cascading can be applied by splitting the unitcell into small unit blocks along the aperture direction. FIG. 5Cillustrates the unit cell in a 3D finite mode, with 200degrees-of-freedom allocated along the horizontal direction (z), 32nodes in the longitudinal direction (x), and 600 nodes in the aperturedirection (y). This results in 120,000 boundary degrees-of-freedom, asshown. In finite 3D device analysis, cascading would be applied in thex-direction to build longer sequences of unit cells.

As shown, the number of degrees-of-freedom increases substantially witheach additional dimension or expansion of the model. With the number ofDOFs per face denoted as N, each B-matrix contains ˜4N² complex-valuedfloating point numbers. The complexity of cascading is O(N³). Table Ilists these numbers for the different analysis types:

Analysis DOFs/side RAM Complexity Time 2D finite 200 4.8 MB 8 · 10⁶  10ms 3D periodic 6400 5 GB 3 · 10¹¹ 5 min 3D finite 120000 2 TB 2 · 10¹⁵25 days

A 2D simulation can be run in a few seconds per frequency point and canbe run on a very modest hardware. A 3D periodic analysis takes a fewhours and requires a heavy desktop with 32-64 GB of RAM. By the samescaling, a 3D finite-structure simulation would require years ofcomputation and several TB of RAM. FEM is particularly computationallydemanding, because calculating frequency responses and other parameterswith a high accuracy requires analyzing several nodes per wavelength,frequently 10, 20, or 30 or more nodes.

A. Modal B-Matrix

The high number of DOFs in FEM is required to suppress numericaldispersion. Basically, this reflects that the FEM shape functions arenot very efficient in describing propagating waves. Assume that thesolution field at the boundaries can be more compactly approximated as asum of N<<N_(DOF) modes u_(i):X=u ₁ y ₁ + . . . +u _(N) y _(N)=[U](y)  (Eq. 21)This can be substituted into Eq. (8). To balance the number of equationswith the number of effective DOFs, the stresses are multiplied withU^(T); this corresponds to change of base functions in FEM in Galerkinformalism.

$\begin{matrix}{{\begin{bmatrix}{U^{T}B_{11}U} & {U^{T}B_{12}U} & {U^{T}B_{13}} \\{U^{T}B_{21}U} & {U^{T}B_{22}U} & {U^{T}B_{23}} \\{B_{31}U} & {B_{32}U} & B_{33}\end{bmatrix}\begin{pmatrix}y_{L} \\y_{R} \\v\end{pmatrix}} = {- \begin{pmatrix}{U^{T}\tau_{L}} \\{U^{T}\tau_{R}} \\{- q}\end{pmatrix}}} & \left( {{Eq}.\mspace{11mu} 22} \right)\end{matrix}$The quantity in brackets is the modal B-matrix B(U). The hierarchicalcascading using modal B-matrices proceeds exactly in the same as withnormal B-matrices. It provides an approximate solution to the simulationproblem within the functions which can be expressed using Eq. (21), butwith complexity O(N³)<<O(N_(DOF) ³).B. Iterative Base Extension

To find bases without knowing the solution, in some implementations, aniterative cascading algorithm may be used. This expands on theimplementation illustrated in FIG. 6A, or direct hierarchical cascading,by iteratively refining guesses for initial core blocks and modes whileprogressively reducing error rates until achieving a final model. A flowchart of an implementation of a method for iterative hierarchicalcascading is illustrated in FIG. 6B. As shown, at step 602, core blockFEM models are identified, and at step 620, an initial guess for thesuitable modes is generated, for example based on 2D periodic solution.

Using the approximate base, the core B-matrices are generated at step622, similar to step 604 discussed above. At steps 606 and 608, thesolution within the approximate base is calculated via the cascadingprocess discussed above, and inverse cascading is applied, if desired,to generate a final approximate solution expressed in terms of theoriginal DOFs.

At step 624, a local error at each unit block boundary is estimated andcompared to a threshold at step 626. In some implementations, the errormay be measured in terms of stress discontinuities across unit blocks,or a difference of displacement to that imposed by approximate boundarystresses. The error vectors across the device form a linear space, whichis partially independent of the original modal base. Selected linearlyindependent components of that error space are included in the modalbase of Eq. (21) at step 628, and the process is repeated. The iterationis continued until sufficient accuracy is reached at step 626 (e.g.accuracy greater than a threshold level, or error rates less than athreshold). As each iteration increases the dimension of the mode base,eventually the process will cover all the original DOFs. Post processingmay then be applied at step 610, as discussed above.

While individual iterations are much faster than cascading using allDOFs, the overall efficiency of the algorithm depends on how rapidly itconverges—on how successfully relevant modes can be added to complementthe mode base.

Examples

The hierarchical cascading algorithm was implemented on the commercialMatlab platform, using a custom mesh generation algorithm and an FEMengine. The mesh generation is based on a modified version of Chew′second Delaunay refinement algorithm; in particular, mechanisms wereintroduced to relax mesh fidelity criteria within thin films. Atriangulated mesh is used in the vicinity of the electrodes: anincreased element density around the electrode corners is highlybeneficial for modeling the charge density distribution, resulting inimproved accuracy of the simulated capacitance. Further away from thesurface the mesh is regular and consists of quadrilateral elements, asshown in FIG. 2B. The simulation speeds are reported for an elderlydesktop PC (CPU i7-2600k, 3.4 GHz, 32 GB RAM), with computationdistributed over four parallel threads.

A. 2D Synchronous Resonator on 42° YX-Cut LiTaO₃

The first example is a synchronous resonator on a 42° YX-cut LiTaO₃substrate, with the following geometry: pitch (electrode-to-electrodedistance) p=1.0 μm, metallization ratio a/p=0.55, aluminum thicknessh_(A1)=160 nm, and acoustic aperture W=40.0 μm. There are N_(t)=121electrodes in the IDT and N_(g)=40 electrodes in each reflector. TheKovacs materials constants were used for the substrate. Material lossesin the electrodes and the substrate were modeled as viscous damping byadding an imaginary component to the elastic constants. Resistive losseswere estimated using bulk conductivity σ_(A1)=3.7·107 S/m.

The substrate was modeled with the C-PML technique, covering H=1.0 μm ofnormal substrate and H_(PML)=2.0 μm of PML with 20 finite elements. Onlya single unit period and the two PML blocks need to be simulated withFEM. The models with quadratic and cubic elements used 6636 and 14 625DOFs, respectively. Note that in conventional FEM this would mean adevice model with 2.7 and 5.9 million variables, respectively. Theachieved simulation speeds were 2.4 and 9.6 seconds per frequency point,respectively. The results were essentially identical; in what follows,those from the quadratic model are shown.

To validate the accuracy of the simulation, the same structure was alsosimulated using a FEM/BEM-based commercial simulation tool FEMSAW2. Thesimulated admittance curves are compared in the graph at the left ofFIG. 7A, with (in this case, the real portion of the admittance (Re(Y))and the absolute admittance (|Y|) computed over the frequency range1,500 MHz-1,800 MHz) of the FEM hierarchical cascading technique and thereference FEM/BEM. The small differences are mostly due to differencesin the modeling of resistivity.

FIG. 7A also illustrates a visualization of the absolute power flow|{right arrow over (P)}| at the selected frequencies shown, evaluatedusing inverse cascading. The white vertical lines mark the boundarybetween reflectors and transducer. Different color scale is used atdifferent frequencies. Losses due to acoustic radiation are manifestedas power flow towards the bottom or through the reflectors. Theresonance frequency (1,964 MHz) shows strong confinement of acousticenergy both laterally and in depth direction. In the middle of thestopband (2,000-2,100 MHz), the most prominent feature is localizedbulk-wave radiation from the transition region between the IDT and thereflectors. Strong synchronous bulk-wave excitation can be seen at 2,150MHz. Similarly, FIG. 7B illustrates a visualization of the dominantshear-horizontal component of the mechanical displacement at theselected frequencies.

B. TC-SAW Simulation on 128° YX-Cut LiNbO₃

The second example demonstrates an advanced SAW structure, a temperaturecompensated SAW (TCSAW) resonator on 128° YX-cut LiNbO₃, with 170 nmthick copper electrodes and a 630 nm thick SiO2 overcoating. Theresonator is synchronous, with pitch p=1 μm, metallization ratio is 0.5,N_(t)=121, N_(g)=40, and W=49.2 μm. Resistive losses were estimatedusing conductivity σ_(Cu)=5.8·107 S/m.

As discussed above, for this substrate the C-PML is unstable in verticaldirection. To demonstrate the impact of the instability, simulationswere carried out using three different substrate mesh configurations,see Table II:

Configuration H H_(PML) Elements μ_(z, max) r C-PML1 1.0p 2.0p 20 80 +80i 0 C-PML2 4.0p 2.0p 40 80 + 80i 0 M-PML 4.0p 36.0p 40 10 + 10i 0.02

Configuration C-PML1 uses the same C-PML mesh as discussed above. C-PML2is otherwise the identical but with more normal substrate between thesurface and the PML; the rationale is to direct radiation from theunstable bottom PML to the stable lateral PMLs. The third one uses anoptimized M-PML. The stretching ratio r≈0.02 seems sufficient tocompletely stabilize the M-PML. However, to suppress reflections thestretching profile μ₃(x₃) must be made more gradual than in the C-PMLapproach. Consequently, a thicker absorbing layer must be used to reacha comparable level of absorption. The side PMLs are implemented usingthe C-PML approach. All PMLs use the attenuation profile from Eq. (6).

The FEM model for a single unit block had 9317-13383 degrees-of-freedom,depending on the mesh configuration; the achieved simulation speedsrespectively varied between 5.1-13.6 seconds per frequency point. Thesimulated electric responses are compared in the admittance overfrequency graphs of FIG. 8. At the main resonance and within thestopband, 1740-1,920 MHz, the results are almost identical. The M-PMLconfiguration appears stable and produces physically sound results atall frequencies. In contrast, C-PML1 is distinctively unstable forfrequencies above 2,000 MHz. C-PML2 performs better than C-PML1. This isdue to better geometric isolation in C-PML2: with more distance betweenthe surface and the bottom PML, most shear waves reach the stable sidePMLs without entering the unstable bottom PML. Nevertheless, C-PML2 alsoshows signs of instability at 2300-2400 MHz. At these frequencies,synchronous excitation from the IDT coincides with the non-convexportion of the slowness curves (s_(x) ⁻¹=2fp).

FIG. 9 shows the Q-factor simulated in the M-PML configuration. TheQ-factor remains high from the resonance to the antiresonance (shown indashed vertical lines). The pronounced radiation losses above thestopband, in the range 1,920-2,030 MHz, arise due to SAWs escapingthrough the reflectors. Above 2,050 MHz, BAW excitation contributessignificantly to losses.

C. 3D Periodic Analysis on 42° YX-Cut LiTaO₃

As an example of 3D periodic analysis with hierarchical cascading, wesimulate the harmonic admittance of an electrode array on 42° YX-cutLiTaO₃, including transversal effects due to finite aperture. Thedimensions of the array are the same as discussed above: p=1.0 μm,h_(A1)=160 nm, and W=40.0 μm. The unit cell is similar to the celldisplayed in FIG. 5 (without side PMLs). For cascading, the unit cellcan be decomposed into the unique unit blocks shown in FIG. 10. The useof the different unit blocks in the mesh is summarized in Table III:

Mode Base Location Occurrence DOFs Active DOFs 1 IDT 129 4638 720 2 Gap1 4575 280 3 Busbar/PML 129 4680 630 4 Gap 1 4575 279 5 Busbar/PML 1294680 606In the above table, occurrence represents the number of block-blockinterfaces using the base; and active DOFs represent the number of modesafter 120 iterations at 1,990 MHz.

As shown, the unit may be deconstructed into five unique unit blocks,consisting of (from left to right in FIG. 10): left busbar & PML block,left busbar-side gap block, left IDT-side gap block, IDT block, rightIDT-side gap block, right busbar-side gap block, and right busbar & PMLblock. For simplicity, transition blocks have been preferred overdiscontinuities across unit block interfaces.

FIG. 11 shows the surface slowness curve for LSAW on 42° YX-cut LiTaO₃,with the solid lines illustrating LSAW under a uniform aluminum lawyerwith thickness-frequency product h f from 0 (solid) to 500 m/s (dashed);and with the red curve and blue cure illustrating bulk acoustic wavesand Rayleigh waves with energy propagating along the crystal surface, insome implementations. The LSAW on metallized surface exhibits non-convexslowness curvature, with the transversal component of power flowopposite to the phase velocity. This results in instability of C-PMLapproach in transversal direction. The side PMLs were implementedcombining M-PML with implementations of the cascading techniquesdiscussed above; and a three-stage M-PML with r=0.02 and a gradedstretching factor was used at each side. The C-PML technique works fineas the bottom PML.

For the limited size device illustrated, the simulation problem iscomputationally feasible using direct hierarchical cascading, but herewe demonstrate the use of iterative cascading. The simulated harmonicadmittance is shown in FIG. 12, which illustrates graphs of admittanceover frequency for example models of a surface acoustic wave devicegenerated via implementations of direct and iterative hierarchicalcascading FEM in two and three dimensions for ease of comparison. The 3Dsimulation shows pronounced radiation at frequencies above theresonance, characteristic to LSAW radiation to busbars. Theinterpretation is the confirmed by the displacement profiles shown inFIG. 13. The vertical lines indicate the boundaries between the PMLs,busbars, and the IDT. Note the asymmetry of the shear verticalcomponent. The longitudinal displacement component u_(x) vanishes due toantisymmetry about the electrode center.

FIG. 14 is an example graph illustrating relative error in estimatedharmonic admittance via iterative hierarchical cascading FEM overiteration count, according to some implementations shows the convergenceof the harmonic admittance at 2,000 MHz. An exponential convergence rateis achieved, however with spontaneous fluctuations in the error rate.This behavior seems characteristic to the iterative cascading, repeatingover wide range of frequencies and from structure to another. Theachieved average simulation speed was about 10-30 min/frequency point,depending on the frequency and on the convergence criterion.

The main advantage of hierarchical cascading in this application is thathigh quality side PMLs could be used, something which would be difficultto achieve in conventional FEM due to the much higher RAM usage of thelatter.

The hierarchical cascading approach has proven an efficient and capabletool for simulating of finite SAW devices with FEM. The electricresponse can be evaluated and loss mechanisms analyzed even in complexlayered SAW structures. The approach has been shown feasible for even 3Dsimulation of finite SAW devices

FIG. 15 is a block diagram of an implementation of a system forhierarchical cascading in FEM simulations and for designing electronicfilters;

FIG. 16 is a block diagram of a wireless telecommunications system,according to some implementations. The systems and methods discussedherein may be applied to designing acoustic wave microwave filters (suchas surface acoustic wave, bulk acoustic wave, film bulk acousticresonator (FBAR), and microelectromechanical system (MEMS) filters)),such as in the 300 MHz to 300 GHz frequency range, particularly in the300 MHz to 10.0 GHz frequency range, and even more particularly in the400-3,500 MHz frequency range. Such microwave filters may be eitherfixed frequency and/or tunable filters (tunable in frequency and/orbandwidth and/or input impedance and/or output impedance), and may beused for single-band or multiple-band bandpass and/or bandstopfiltering. Such AW microwave filters are advantageous in applicationsthat have demanding electrical and/or environmental performancerequirements and/or severe cost/size constraints, such as those found inthe radio frequency (RF) front ends of mobile communications devices,including cellphones, smartphones, laptop computers, tablet computers,etc. or the RF front ends of fixed-location or fixed-path communicationsdevices, including M2M devices, wireless base stations, satellitecommunications systems, etc.

Example AW microwave filters described herein exhibit a frequencyresponse with a single passband, which is particularly useful intelecommunication system duplexers. For example, with reference to FIG.16, a telecommunications system 1610 for use in a mobile communicationsdevice may include a transceiver 1612 capable of transmitting andreceiving wireless signals, and a controller/processor 1614 capable ofcontrolling the functions of the transceiver 1612. The transceiver 1612generally comprises a broadband antenna 1616, a duplexer 1618 having atransmit filter 1624 and a receive filter 1626, a transmitter 1620coupled to the antenna 1616 via the transmit filter 1624 of the duplexer1618, and a receiver 1622 coupled to the antenna 1616 via the receivefilter 1626 of the duplexer 1618.

The transmitter 1620 includes an upconverter 1628 configured forconverting a baseband signal provided by the controller/processor 1614to an RF signal, a variable gain amplifier (VGA) 1630 configured foramplifying the RF signal, a bandpass filter 1632 configured foroutputting the RF signal within an operating frequency band selected bythe controller/processor 1614, and a power amplifier 1634 configured foramplifying the filtered RF signal, which is then provided to the antenna1616 via the transmit filter 1624 of the duplexer 1618.

The receiver 1622 includes a notch or stopband filter 1636 configuredfor rejecting signal interference from the RF signal input from theantenna 1616 and transmitter 1620 via the receiver filter 1626, a lownoise amplifier (LNA) 1638 configured for amplifying the RF signal fromthe stop band filter 1636 with a relatively low noise, a bandpass filter1640 configured for outputting the amplified RF signal within anoperating frequency band selected by the controller/processor 1614, anda downconverter 1642 configured for down-converting the RF signal to abaseband signal that is provided to the controller/processor 1614.Alternatively, the function of rejecting signal interference performedby the stop-band filter 1636 can instead or also be performed by theduplexer 1618. And/or, the power amplifier 1634 of the transmitter 1620can be designed to reduce the signal interference to the receiver 1622.

It should be appreciated that the block diagram illustrated in FIG. 16is functional in nature, and that several functions can be performed byone electronic component or one function can be performed by severalelectronic components. For example, the functions performed by the upconverter 1628, VGA 1630, bandpass filter 1640, downconverter 1642, andcontroller/processor 1614 are oftentimes performed by a singletransceiver chip or device. The function of the bandpass filter 1632 canbe performed by the power amplifier 1634 and the transmit filter 1624 ofthe duplexer 1618.

The exemplary technique described herein is used to design acousticmicrowave filters for the RF front-end, comprised of the duplexer 1618,transmitter 1620, and receiver 1622, of the telecommunications system1610, and in particular the transmit filter 1624 of the duplexer 1618,although the same technique can be used to design acoustic microwavefilters for the receive filter 1626 of the duplexer 1618 and for otherRF filters in the wireless transceiver 1612.

FIG. 17 is a flow diagram illustrating the design and construction of afilter using the FEM hierarchical cascading technique, according to someimplementations. First, the filter requirements, which comprise thefrequency response requirements (including passband, return loss,insertion loss, rejection, linearity, noise figure, input and outputimpedances, etc.), as well as size and cost requirements, andenvironmental requirements, such as operating temperature range,vibration, failure rate, etc., are defined to satisfy the application ofthe filter (step 1702).

Next, the structural types of circuit elements to be used in the AWfilter are selected; for example, the structural type of AW resonatorsand/or coupling elements (SAW, BAW, FBAR, MEMS, etc.) and the types ofinductors, capacitors, and switches, along with the materials to be usedto fabricate these circuit elements, including the packaging andassembly techniques for fabricating the filter, are selected (step1704). For example, as discussed above, SAW resonators may be selected,which may be fabricated by disposing IDTs on a piezoelectric substrate,such as crystalline Quartz, Lithium Niobate (LiNbO₃), Lithium Tantalate(LiTaO₃) crystals or BAW (including FBAR) resonators or MEMS resonators.In the particular example described herein, the selected circuit elementtypes are SAW resonators and capacitors constructed on a substratecomposed of 42-degree X Y cut LiTaO₃.

Then, a filter circuit topology is selected (step 1706). For example,the selected filter circuit topology may be an Nth-order ladder topology(in this case, N=6 meaning the number of resonators equals 6). Nth orderladder topologies are described in U.S. Pat. Nos. 8,751,993 and8,701,065 and U.S. patent application Ser. No. 14/941,451, entitled“Acoustic Wave Filter with Enhanced Rejection,” which are all expresslyincorporated herein by reference. Other filter circuit topologies, suchas in-line non-resonant-node, or in-line, or in-line with crosscouplings, or in-line.

Then, initial physical models of the filter's AW components are defined(or modified), e.g., by selecting a material, one or more of a number offinger pairs, aperture size, mark-to-pitch ratio, and/or transducermetal thickness (step 1708), and the physical models of the AWcomponents are simulated using the FEM hierarchical cascading techniqueto determine their frequency-dependent electrical characteristics (step1710). Next, these electrical characteristics of the AW components areincorporated into a circuit model of the entire filter network (step1712), and the circuit model of the filter network is simulated(optionally optimizing non-AW component parameters) to determine thefilter's frequency characteristics (step 1714). The simulated frequencyresponse of the AW filter is then compared to the frequency responserequirements defined at step 1702 (step 1716). If the simulatedfrequency response does not satisfy the frequency response requirements,the process returns to step 1708, where the physical model of the AW ismodified. If the simulated frequency response does satisfy the frequencyresponse requirements (step 1702), an actual acoustic filter isconstructed based on the most recent physical models of the AWcomponents (step 1714). Preferably, the circuit element values of theactual acoustic filter will match the corresponding circuit elementvalues in the most recent optimized filter circuit design.

Although the FEM hierarchical cascading technique has been disclosedherein as being applied to SAW structures having strict periodicity, itshould be appreciated that the FEM hierarchical cascading technique canbe applied to devices having breaks in periodicity, such as “hiccup”resonators or devices with “accordion sections.” In the case of suchdevices, the FEM hierarchical cascading technique can be applied to thestrictly periodic structures, whereas “one-off” cells or small number ofaperiodic cells can be inserted between the periodic sections. Also, itshould be clear to a person skilled in the art that the term “SAW,” asused herein, includes all types of acoustic waves, such asquasi-Rayleigh waves, “leaky” SAW, Surface Transverse Waves, STW, Lambmodes, etc.—that is, all types of acoustic waves with propagation mainlynear the surface of, or in a layer of limited depth, for whichcomponents radiated into the bulk represent undesirable “second-order”effects.

Referring to FIG. 18, a computerized filter design system 1,800 may beused to simulate an AW structure and an AW filter using the designprocedures discussed herein. The computerized filter design system 1,800generally comprises a user interface 1802 configured for receivinginformation and data from a user (e.g., parameter values defining thephysical model of the AW structure and AW filter requirements) andoutputting frequency-dependent characteristics of the AW structure andfilter to the user; a memory 1804 configured for storing a filter designprogram 1808 (which may take the form of software instructions, whichmay include, but are not limited to, routines, programs, objects,components, data structures, procedures, modules, functions, and thelike that perform particular functions or implement particular abstractdata types), as well as the information and data input from the user viathe user interface 1802; and a processor 1806 configured for executingthe simulation software program.

The simulation software program 1808 is divided into sub-programs, inparticular, a conventional FEM program 1810 (which can be used tocompute characteristics of the core blocks and PML absorber blocks); ahierarchical cascading program 1812 (which can be used to partition thephysical model, identify core blocks, compute the characteristics of thecore blocks, remove DOFs from core blocks, define types of unit blocks,determine hierarchical cascading pattern, identify and cascade sets ofadjacent unit blocks, recognize a single unit block subsuming alloriginal unit blocks, terminate the single block with absorber blocks,compute characteristics of absorber blocks, cascade the single subsumingblock with the absorber blocks, and determine the frequency-dependentelectrical characteristics of the entire terminated AW structure; and aconventional filter optimizer 1814 (which can be used to optimize andsimulate the circuit model of the filter network).

Accordingly, the systems and methods discussed herein provide forhierarchical cascading in FEM simulations of SAW devices, which offersdrastically reduced memory consumption and simulation times. In someimplementations, iterative hierarchical cascading may also be applied tothree-dimensional simulations of SAW devices, which may otherwise be toocomplex for FEM simulations due to the high number of cross-sectionaldegrees-of-freedom involved

In a first aspect, the present disclosure is directed to a method ofgenerating an acoustic wave device. The method includes (a)partitioning, by a computing system, a physical model of an acousticwave device into a plurality of core unit blocks. The method alsoincludes (b) computing, by the computing system, characteristics for afirst core unit block of the plurality of core unit blocks according toa modal matrix based on a first set of basis values. The method alsoincludes (c) calculating, by the computing system, a single blockrepresenting the physical model of the acoustic wave device based on thecomputed characteristics for the first core unit block of the pluralityof core unit blocks and derived characteristics for each other core unitblock of the plurality of core unit blocks. The method also includes (d)determining, by the computing system, that one or more local errors ateach boundary of the plurality of core unit blocks exceeds a threshold.The method also includes (e) responsive to the determination, repeatingsteps (b)-(d) with an adjusted modal matrix based on a second set ofbasis values, the second set of basis values comprising at least oneindependent component of an error vector associated with the one or morelocal errors. The method also includes (f) comparing, by the computingsystem, a frequency response of the calculated single block representingthe physical model of the acoustic wave device to a set of frequencyresponse requirements, responsive to determining that the one or morelocal errors at each boundary of the plurality of core unit blocks donot exceed the threshold. The method also includes (g) generating, bythe computing system, a set of specifications for the acoustic wavedevice based on the comparison, the set of specifications serving as aninput to a manufacturing process.

In some implementations, the method includes deriving, by the computingsystem, characteristics for each other core unit block of the pluralityof core unit blocks from the computed characteristics for the first coreunit block; and combining, by the computing system, the first core unitblock and each other core unit block into the single block such that thesingle block subsumes the first core unit block and each other core unitblock. In a further implementation, the method includes hierarchicallycascading sets of adjacent unit blocks into the single block. In a stillfurther implementation, the method includes (h) combining sets ofadjacent unit blocks at a current hierarchical level to create cascadedunit blocks at a next hierarchical level; and (i) repeating step (h) forsets of adjacent unit blocks for the next hierarchical level until thesingle block is created, wherein each of the unit blocks is either acore unit block or a previously cascaded unit block. In a yet stillfurther implementation, any of the unit blocks that are not combined atthe current hierarchical level are transferred from the currenthierarchical level to the next hierarchical level. In a still yetfurther implementation, a first unit block has previously computedcharacteristics, and at least one other of the unit blocks is physicallyand electrically identical to the first unit block, and the methodfurther includes referencing the first unit block to assume thepreviously computed characteristics for the at least one other unitblock when combining the sets of adjacent unit blocks at the currenthierarchical level.

In some implementations, the method includes generating an A-matrixhaving left-side boundary degrees of freedom (DOFs), right-side boundaryDOFs, and internal DOFs; and removing the internal DOFs from theA-matrix to generate a B-matrix comprising only the left-side boundaryDOFs and the right-side boundary DOFs. In a further implementation, thecharacteristics of each other core unit block of the plurality of coreunit blocks are derived from the B-matrix of the first core unit block.In a still further implementation, the method includes cascading a firstset of adjacent unit blocks into a first cascaded unit block by:combining B-matrices of the respective adjacent unit blocks of the firstset into a first C-matrix having left-side boundary DOFs correspondingto the left-side boundary DOFs of a left one of the adjacent unitblocks, right-side boundary DOFs corresponding to a right one of theadjacent unit blocks, and internal DOFs corresponding to at least oneshared edge between the adjacent unit blocks; and reducing the firstC-matrix by removing the internal DOFs from the first C-matrix to afirst new cascaded B-matrix of a first cascaded unit block comprisingonly left-side boundary DOFs and right-side boundary DOFs.

In some implementations, the method includes identifying the one or morelocal errors as stress discontinuities across boundaries between unitblocks. In some implementations, the method includes identifying the oneor more local errors as differences of displacement to that imposed byapproximate boundary stresses between unit blocks. In someimplementations, the computed characteristics for the first core unitblock comprise acoustic and electric fields. In some implementations,all of the core unit blocks are physically identical to each other. Inother implementations, at least two of the core unit blocks arephysically different from each other.

In another aspect, the present disclosure is directed to a filter designsystem. The system includes a processor; an interface coupled to theprocessor; and memory storing a hierarchical cascading program.Execution of the hierarchical cascading program by the processor causesthe filter design system to perform actions comprising: (a) partitioninga physical model of an acoustic wave device into a plurality of coreunit blocks; (b) computing characteristics for a first core unit blockof the plurality of core unit blocks according to a modal matrix basedon a first set of basis values; (c) calculating a single blockrepresenting the physical model of the acoustic wave device based on thecomputed characteristics for the first core unit block of the pluralityof core unit blocks and derived characteristics for each other core unitblock of the plurality of core unit blocks; (d) determining that one ormore local errors at each boundary of the plurality of core unit blocksexceeds a threshold; (e) responsive to the determination, repeatingsteps (b)-(d) with an adjusted modal matrix based on a second set ofbasis values, the second set of basis values comprising at least oneindependent component of an error vector associated with the one or morelocal errors; (f) comparing a frequency response of the calculatedsingle block representing the physical model of the acoustic wave deviceto a set of frequency response requirements, responsive to determiningthat the one or more local errors at each boundary of the plurality ofcore unit blocks do not exceed the threshold; and (g) generating a setof specifications for the acoustic wave device based on the comparison,the set of specifications serving as an input to a manufacturingprocess.

In some implementations, execution of the hierarchical cascading programfurther causes the filter design system to: derive characteristics foreach other core unit block of the plurality of core unit blocks from thecomputed characteristics for the first core unit block; and combine thefirst core unit block and each other core unit block into the singleblock such that the single block subsumes the first core unit block andeach other core unit block. In some implementations, In someimplementations, execution of the hierarchical cascading program furthercauses the filter design system to hierarchically cascade sets ofadjacent unit blocks into the single block. In a further implementation,execution of the hierarchical cascading program further causes thefilter design system to: (h) combine sets of adjacent unit blocks at acurrent hierarchical level to create cascaded unit blocks at a nexthierarchical level; and (i) repeat step (h) for sets of adjacent unitblocks for the next hierarchical level until the single block iscreated, wherein each of the unit blocks is either a core unit block ora previously cascaded unit block.

In some implementations, execution of the hierarchical cascading programfurther causes the filter design system to identify the one or morelocal errors as stress discontinuities across boundaries between unitblocks. In some implementations, execution of the hierarchical cascadingprogram further causes the filter design system to identify the one ormore local errors as differences of displacement to that imposed byapproximate boundary stresses between unit blocks.

Implementations of the subject matter and the operations described inthis specification can be implemented in digital electronic circuitry,or in computer software, firmware, or hardware, including the structuresdisclosed in this specification and their structural equivalents, or incombinations of one or more of them. Implementations of the subjectmatter described in this specification can be implemented as one or morecomputer programs, i.e., one or more modules of computer programinstructions, encoded on one or more computer storage medium forexecution by, or to control the operation of, data processing apparatus.Alternatively or in addition, the program instructions can be encoded onan artificially-generated propagated signal, e.g., a machine-generatedelectrical, optical, or electromagnetic signal, that is generated toencode information for transmission to suitable receiver apparatus forexecution by a data processing apparatus. A computer storage medium canbe, or be included in, a computer-readable storage device, acomputer-readable storage substrate, a random or serial access memoryarray or device, or a combination of one or more of them. Moreover,while a computer storage medium is not a propagated signal, a computerstorage medium can be a source or destination of computer programinstructions encoded in an artificially-generated propagated signal. Thecomputer storage medium can also be, or be included in, one or moreseparate components or media (e.g., multiple CDs, disks, or otherstorage devices). Accordingly, the computer storage medium may betangible.

The operations described in this specification can be implemented asoperations performed by a data processing apparatus on data stored onone or more computer-readable storage devices or received from othersources.

The term “client or “server” include all kinds of apparatus, devices,and machines for processing data, such as a programmable processor, acomputer, a system on a chip, or multiple ones, or combinations, of theforegoing. The apparatus can include special purpose logic circuitry,e.g., an FPGA (field programmable gate array) or an ASIC(application-specific integrated circuit). The apparatus can alsoinclude, in addition to hardware, code that creates an executionenvironment for the computer program in question, e.g., code thatconstitutes processor firmware, a protocol stack, a database managementsystem, an operating system, a cross-platform runtime environment, avirtual machine, or a combination of one or more of them. The apparatusand execution environment can realize various different computing modelinfrastructures, such as web services, distributed computing and gridcomputing infrastructures.

A computer program (also known as a program, software, softwareapplication, script, or code) can be written in any form of programminglanguage, including compiled or interpreted languages, declarative orprocedural languages, and it can be deployed in any form, including as astand-alone program or as a module, component, subroutine, object, orother unit suitable for use in a computing environment. A computerprogram may, but need not, correspond to a file in a file system. Aprogram can be stored in a portion of a file that holds other programsor data (e.g., one or more scripts stored in a markup languagedocument), in a single file dedicated to the program in question, or inmultiple coordinated files (e.g., files that store one or more modules,sub-programs, or portions of code). A computer program can be deployedto be executed on one computer or on multiple computers that are locatedat one site or distributed across multiple sites and interconnected by acommunication network.

The processes and logic flows described in this specification can beperformed by one or more programmable processors executing one or morecomputer programs to perform actions by operating on input data andgenerating output. The processes and logic flows can also be performedby, and apparatus can also be implemented as, special purpose logiccircuitry, e.g., an FPGA (field programmable gate array) or an ASIC(application specific integrated circuit).

Processors suitable for the execution of a computer program include bothgeneral and special purpose microprocessors, and any one or moreprocessors of any kind of digital computer. Generally, a processor willreceive instructions and data from a read-only memory or a random accessmemory or both. The essential elements of a computer are a processor forperforming actions in accordance with instructions and one or morememory devices for storing instructions and data. Generally, a computerwill also include, or be operatively coupled to receive data from ortransfer data to, or both, one or more mass storage devices for storingdata, e.g., magnetic, magneto-optical disks, or optical disks. However,a computer need not have such devices. Moreover, a computer can beembedded in another device, e.g., a mobile telephone, a personal digitalassistant (PDA), a mobile audio or video player, a game console, aGlobal Positioning System (GPS) receiver, or a portable storage device(e.g., a universal serial bus (USB) flash drive), to name just a few.Devices suitable for storing computer program instructions and datainclude all forms of non-volatile memory, media and memory devices,including semiconductor memory devices, e.g., EPROM, EEPROM, and flashmemory devices; magnetic disks, e.g., internal hard disks or removabledisks; magneto-optical disks; and CD-ROM and DVD-ROM disks. Theprocessor and the memory can be supplemented by, or incorporated in,special purpose logic circuitry.

To provide for interaction with a user, implementations of the subjectmatter described in this specification can be implemented on a computerhaving a display device, e.g., a CRT (cathode ray tube), LCD (liquidcrystal display), OLED (organic light emitting diode), TFT (thin-filmtransistor), plasma, other flexible configuration, or any other monitorfor displaying information to the user and a keyboard, a pointingdevice, e.g., a mouse, trackball, etc., or a touch screen, touch pad,etc., by which the user can provide input to the computer. Other kindsof devices can be used to provide for interaction with a user as well;feedback provided to the user can be any form of sensory feedback, e.g.,visual feedback, auditory feedback, or tactile feedback; and input fromthe user can be received in any form, including acoustic, speech, ortactile input. In addition, a computer can interact with a user bysending documents to and receiving documents from a device that is usedby the user; by sending webpages to a web browser on a user's clientdevice in response to requests received from the web browser.

Implementations of the subject matter described in this specificationcan be implemented in a computing system that includes a back-endcomponent, e.g., as a data server, or that includes a middlewarecomponent, e.g., an application server, or that includes a front-endcomponent, e.g., a client computer having a graphical user interface ora Web browser through which a user can interact with an implementationof the subject matter described in this specification, or anycombination of one or more such back-end, middleware, or front-endcomponents. The components of the system can be interconnected by anyform or medium of digital data communication, e.g., a communicationnetwork. Communication networks may include a local area network (“LAN”)and a wide area network (“WAN”), an inter-network (e.g., the Internet),and peer-to-peer networks (e.g., ad hoc peer-to-peer networks).

While this specification contains many specific implementation details,these should not be construed as limitations on the scope of anyinventions or of what may be claimed, but rather as descriptions offeatures specific to particular implementations of particularinventions. Certain features that are described in this specification inthe context of separate implementations can also be implemented incombination in a single implementation. Conversely, various featuresthat are described in the context of a single implementation can also beimplemented in multiple implementations separately or in any suitablesubcombination. Moreover, although features may be described above asacting in certain combinations and even initially claimed as such, oneor more features from a claimed combination can in some cases be excisedfrom the combination, and the claimed combination may be directed to asubcombination or variation of a subcombination.

Similarly, while operations are depicted in the drawings in a particularorder, this should not be understood as requiring that such operationsbe performed in the particular order shown or in sequential order, orthat all illustrated operations be performed, to achieve desirableresults. In certain circumstances, multitasking and parallel processingmay be advantageous. Moreover, the separation of various systemcomponents in the implementations described above should not beunderstood as requiring such separation in all implementations, and itshould be understood that the described program components and systemscan generally be integrated together in a single software product orpackaged into multiple software products.

Thus, particular implementations of the subject matter have beendescribed. Other implementations are within the scope of the followingclaims. In some cases, the actions recited in the claims can beperformed in a different order and still achieve desirable results. Inaddition, the processes depicted in the accompanying figures do notnecessarily require the particular order shown, or sequential order, toachieve desirable results. In certain implementations, multitasking orparallel processing may be utilized.

What is claimed:
 1. A method of generating an acoustic wave device,comprising: (a) partitioning, by a computing system, a physical model ofan acoustic wave device into a plurality of core unit blocks; (b)computing, by the computing system, characteristics for a first coreunit block of the plurality of core unit blocks according to a modalmatrix based on a first set of basis values; (c) calculating, by thecomputing system, a single block representing the physical model of theacoustic wave device based on the computed characteristics for the firstcore unit block of the plurality of core unit blocks and derivedcharacteristics for each other core unit block of the plurality of coreunit blocks; (d) determining, by the computing system, that one or morelocal errors at each boundary of the plurality of core unit blocksexceeds a threshold; (e) responsive to the determination, repeatingsteps (b)-(d) with an adjusted modal matrix based on a second set ofbasis values, the second set of basis values comprising at least oneindependent component of an error vector associated with the one or morelocal errors; (f) comparing, by the computing system, a frequencyresponse of the calculated single block representing the physical modelof the acoustic wave device to a set of frequency response requirements;and (g) generating, by the computing system, a set of specifications forthe acoustic wave device based on the comparison, the set ofspecifications serving as an input to a manufacturing process.
 2. Themethod of claim 1, wherein calculating the single block furthercomprises: deriving, by the computing system, characteristics for eachother core unit block of the plurality of core unit blocks from thecomputed characteristics for the first core unit block; and combining,by the computing system, the first core unit block and each other coreunit block into the single block such that the single block subsumes thefirst core unit block and each other core unit block.
 3. The method ofclaim 2, wherein combining the first core unit block and each other coreunit block into the single block further comprises hierarchicallycascading sets of adjacent unit blocks into the single block.
 4. Themethod of claim 3, wherein hierarchically cascading sets of adjacentunit blocks into the single block further comprises: (h) combining setsof adjacent unit blocks at a current hierarchical level to createcascaded unit blocks at a next hierarchical level; and (i) repeatingstep (h) for sets of adjacent unit blocks for the next hierarchicallevel until the single block is created, wherein each of the unit blocksis either a core unit block or a previously cascaded unit block.
 5. Themethod of claim 4, wherein any of the unit blocks that are not combinedat the current hierarchical level are transferred from the currenthierarchical level to the next hierarchical level.
 6. The method ofclaim 5, wherein a first unit block has previously computedcharacteristics, and at least one other of the unit blocks is physicallyand electrically identical to the first unit block, the method furthercomprising referencing the first unit block to assume the previouslycomputed characteristics for the at least one other unit block whencombining the sets of adjacent unit blocks at the current hierarchicallevel.
 7. The method of claim 1, wherein computing characteristics forthe first core unit block further comprises generating an A-matrixhaving left-side boundary degrees of freedom (DOFs), right-side boundaryDOFs, and internal DOFs; and removing the internal DOFs from theA-matrix to generate a B-matrix comprising only the left-side boundaryDOFs and the right-side boundary DOFs.
 8. The method of claim 7, whereinthe characteristics of each other core unit block of the plurality ofcore unit blocks are derived from the B-matrix of the first core unitblock.
 9. The method of claim 8, wherein hierarchically cascading setsof adjacent unit blocks into the single block further comprises:cascading a first set of adjacent unit blocks into a first cascaded unitblock by: combining B-matrices of the respective adjacent unit blocks ofthe first set into a first C-matrix having left-side boundary DOFscorresponding to the left-side boundary DOFs of a left one of theadjacent unit blocks, right-side boundary DOFs corresponding to a rightone of the adjacent unit blocks, and internal DOFs corresponding to atleast one shared edge between the adjacent unit blocks, and reducing thefirst C-matrix by removing the internal DOFs from the first C-matrix toa first new cascaded B-matrix of a first cascaded unit block comprisingonly left-side boundary DOFs and right-side boundary DOFs.
 10. Themethod of claim 1, further comprising identifying the one or more localerrors as stress discontinuities across boundaries between unit blocks.11. The method of claim 1, further comprising identifying the one ormore local errors as differences of displacement to that imposed byapproximate boundary stresses between unit blocks.
 12. The method ofclaim 1, wherein the computed characteristics for the first core unitblock comprise acoustic and electric fields.
 13. The method of claim 1,wherein all of the core unit blocks are physically identical to eachother.
 14. The method of claim 1, wherein at least two of the core unitblocks are physically different from each other.
 15. A filter designsystem, comprising: a processor; an interface coupled to the processor;and memory storing a hierarchical cascading program that, when executedby the processor, causes the filter design system to perform actionscomprising: (a) partitioning a physical model of an acoustic wave deviceinto a plurality of core unit blocks; (b) computing characteristics fora first core unit block of the plurality of core unit blocks accordingto a modal matrix based on a first set of basis values; (c) calculatinga single block representing the physical model of the acoustic wavedevice based on the computed characteristics for the first core unitblock of the plurality of core unit blocks and derived characteristicsfor each other core unit block of the plurality of core unit blocks; (d)determining that one or more local errors at each boundary of theplurality of core unit blocks exceeds a threshold; (e) responsive to thedetermination, repeating steps (b)-(d) with an adjusted modal matrixbased on a second set of basis values, the second set of basis valuescomprising at least one independent component of an error vectorassociated with the one or more local errors; (f) comparing a frequencyresponse of the calculated single block representing the physical modelof the acoustic wave device to a set of frequency response requirements;and (g) generating a set of specifications for the acoustic wave devicebased on the comparison, the set of specifications serving as an inputto a manufacturing process.
 16. The filter design system of claim 15,wherein execution of the hierarchical cascading program further causesthe filter design system to: derive characteristics for each other coreunit block of the plurality of core unit blocks from the computedcharacteristics for the first core unit block; and combine the firstcore unit block and each other core unit block into the single blocksuch that the single block subsumes the first core unit block and eachother core unit block.
 17. The filter design system of claim 15, whereinexecution of the hierarchical cascading program further causes thefilter design system to hierarchically cascade sets of adjacent unitblocks into the single block.
 18. The filter design system of claim 17,wherein execution of the hierarchical cascading program further causesthe filter design system to: (h) combine sets of adjacent unit blocks ata current hierarchical level to create cascaded unit blocks at a nexthierarchical level; and (i) repeat step (h) for sets of adjacent unitblocks for the next hierarchical level until the single block iscreated, wherein each of the unit blocks is either a core unit block ora previously cascaded unit block.
 19. The filter design system of claim15, wherein execution of the hierarchical cascading program furthercauses the filter design system to identify the one or more local errorsas stress discontinuities across boundaries between unit blocks.
 20. Thefilter design system of claim 15, wherein execution of the hierarchicalcascading program further causes the filter design system to identifythe one or more local errors as differences of displacement to thatimposed by approximate boundary stresses between unit blocks.